OF KOTATIXC4 LIQUID CYLIXDEKS. 
73 
At points on the boundary the minimum value of 4ah j | is 4o ; hence, provided 
a" > (a — hy, a convergent expansion for ^ is 
a — h 
[a h)-q -j- '2 y^/{ah) 77 ^ 1 ^ 
a — h 
f ^ 1 a - h _ T (a ~ I f 
\ ‘' ~ abr)^ “ 
a — y/b 1 
~^a-[-y'b'^ y/{((b)T] 
1 
4 
(aljfhf 
From this expansion we oljtain at once 
] 
• • (30). 
V, = - 
2tvp 
+ y/& 
( v/“T \/%') 
A = 
TT 
a = 0 
(31) 
W e can obtain Yq in series at once; if Ave require its value in finite teims we may 
proceed as follows :— 
From equation (30), 
Yn = 
^p\ 
= 7rp) 
7, \ v/(rt&) V 
>) vl' 
— A ‘ + • • • + same function of A 
('<«)■ ■'/ / 
c/t^ 
-h same function of ^ 
= Ah i - V + 2 + 4 (a - b) 
3^/(«?>) 7 ^ + a 4- 4 (a — 1>)| 
+ same function of ^. . . (32). 
The results obtained agree with the knoAvn resnlts if y/a and y/h are taken of the 
same sign. There is a second solution, obtained by changing the sign of one of these 
roots, and this corresponds to a mathematical ellipse of Avhich one axis is negative. 
The full significance of this Avill appear' later. 
+ (7A,'»U 
Expansion in Powers of a Parameter. 
§ 8 . When the equation to the .surface is of a degree higher than the second, it will 
not, in general, be possible to obtain a solution in finite terms of the form of equation (29). 
Suppose, hoAvever, that the surface forms one of a family of surfaces, the family being 
descrifjed by the variation of a parameter c, and let this family be chosen so that the 
surface c = 0 is one for which the complete solution is known. Then as we know 
the value of ^ when c = 0 , Ave sliall assume that it Avill be possible to find the general 
value of ^ in a series of ascending poAvers of c, and the equations determining the 
various coefficients of c will have a unique solution. 
VOL. cc.— A. 
L 
